t-Test: Perform a statistical test

The goals of this section are as follows:

1. To choose the correct statistical procedure to test our hypothesis.

2. To correctly interpret the chosen procedure.

3. To choose the correct tabled value (aka critical value) with which to compare your result.

Warning: The Expert says: this section will be difficult and may make little sense unless you have finished both the “Look at Your Data” section and the “State Your Expectations” section. Additionally, information from those sections will be used later to evaluate your work. I strongly recommend completing these sections if you haven’t done so.  To test our hypothesis that children in Samoa are further from their mother than children in Belize, we can perform a statistical test. The test you choose is strongly influenced by what your data looks like and what you want to say about your data (i.e. your hypothesis).

Which of the following tests would you use to test our hypothesis?

Pearson Correlation Coefficient (r)

Incorrect. While correlation is a useful tool for describing the relationship between 2 variables it does not serve as a test of our hypothesis. If we were to calculate the correlation between scores for Belize and Samoa, we could get an r value. However, this value would have no real meaning as it would describe the linear relationship between the scores, but, provide no information regarding similarities or differences between the two cultures. In general, correlations are only appropriate when we are interested only in the relationship of two dissimilar variables (e.g. income and age) rather than two very similar variables.

Regression Analysis

Incorrect. Regression analysis can be used to derive an equation from which we can predict scores. Regression is however, inappropriate in this case for a variety of reasons. First, regression analysis is not usually used to test hypotheses about the differences between groups (though advanced uses of regression analysis do allow for this option). Additionally, regression is inappropriate for our data because it makes little sense to base predictions about Samoa on scores from our Belize sample.

t Test for Independent Means

Correct! The t test for independent means is used to compare means derived from unrelated samples. We have 2 independent groups here, children from Belize and children from Samoa. To compare these means we can use a t-test for independent means.

Paired t-test

Incorrect. A paired t-test (aka t-test for correlated means) is appropriate for data which is paired, matched, or before-after. In this case, we have 2 groups which we have no reason to believe constitute a “pair.” Because our groups do not constitute a pair, we cannot use the paired t-test. If, for example, we had collected data from Belize (24 families) and for each family, we had a family in Samoa that was exactly the same in terms of income, age of mother, age of child, size of community, etc. then, we could argue for a “paired” test. In this case, we have taken each family from 1 culture (Belize) and matched them with a family very similar to them in the other culture (Samoa). Since for each pair of families, we have made sure that they are the same, the only difference between the families can be attributed to culture. In general, the paired designed is preferred but, it is often impractical.

Ask the Expert

When deciding on an appropriate statistical test, it is essential to pay attention to your data and your hypotheses.
Try asking yourself the following questions:

1. Do I have groups of data or continuous scores on two variables? If you have groups you need to use a test that looks at groups such as an independent t-test. If you want to compare the scores on two different variables, correlation/regression is your best bet.

2. Do I have a hypothesis? T tests can be used to test hypotheses when we have a categorical independent variable (i.e. culture) whereas correlation/regression is usually not used in this manner (these techniques can be used for categorical data but are more often used on continuous variables).

3. If you have groups, are they independent or paired? Independent groups are unrelated to each other. Paired scores are usually either before-after or matched pairs type designs.

Before we do the test, we need to state a formal hypothesis. This hypothesis needs to be stated in terms of the Null and Alternative hypotheses, .

Recall that we had decided informally that our hypothesis (expectation) was as follows: Children in Samoa are further from their mothers than children in Belize.

How would we state this in terms of our Null and Alternative hypotheses?

A.
B.
C.
D.
E.

A

Incorrect. This hypothesis is correct in form and could be a valid hypothesis for the data if our informal hypothesis was simply that both groups were not equal. This what we have stated in the alternative hypothesis here. However, we want to determine whether children in Belize are closer to their mothers than children in Samoa; this should be the alternative hypothesis.

If we were only interested stating that the two groups were different (i.e. a two-tailed test), the hypothesis you have chosen would be the correct hypothesis.

B

This hypothesis is incorrect.
1) Our null hypothesis is never “does not equal.” There is always some form of equal sign (either equal, more than or equal to, or less than or equal to) in our null hypothesis. This is because our t-test is testing how different (or unequal) or means are with the initial assumption that the two groups are equal.
2) Our alternative hypothesis (what we want to say, i.e. the informal hypothesis we stated before) does not reflect our hypothesis. The alternative hypothesis should be equivalent to the hypothesis we stated previous (that children in Belize were closer to their mothers than children in Samoa). Our hypothesis here states that the 2 groups are equal; this is not what we want to be able to conclude about the data.

C

Incorrect. This hypothesis is incorrect for a couple of reasons.
1) Our Null and Alternative Hypotheses do not encompass all possibilities. Our groups can either be equal or not equal (not equal can be broken down to less than, or greater than). Here our Null hypothesis is that Belize and Samoa are equal, thus our alternative hypothesis must encompass all other options (not equal in both directions), however, our alternative hypothesis states only that the Belize children are further from their mothers than children in Samoa. There is no consideration of the possibility that Belize children may be closer than children in Samoa.
2) Our alternative hypothesis states that Belize children are further from their mothers than children in Samoa. We want to state that Belize children are closer to their mothers than children in Samoa.

D

Incorrect. Our alternative hypothesis states that Belize children are further from their mothers than children in Samoa. We want to state that Belize children are closer to their mothers than children in Samoa.

E

Correct!

Our alternative hypothesis states that Belize children are closer to their mothers than children in Samoa. This is exactly what we want to state.

Ask the Expert

Stating a formal hypothesis can be done in variety of ways. One of the easier ways in which to create your hypotheses is to 1) decide what you want to conclude based on the literature (what we previously called our informal hypothesis) and then 2) Convert this to a mathematical equation based (i.e. one group Belize < Samoa). 3) Convert our culture names to the symbol representing them (either mu1 or mu2). This is our alternative hypothesis.

Now to state our null hypothesis, we simply include all other possibilities. If our alternative hypothesis is that one group is < than the other, then the null would have to include all other possibilities, i.e. greater than or equal to.

Drawing a Conclusion

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